Reproducing Polynomials with B-Splines

A B-Spline of order $N$ is known to be able to reproduce any polynomial up to order $N$:

$\sum _{n\in \mathbb {Z} }c_{m,n}\beta _{N}(t-n)=t^{m}$

In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order $N$. This is needed for certain applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework where where any kernel $\varphi$ reproducing polynomials, that is, satisfying the Strang-Fix conditions, can be used. However, among all possible kernels, the B-Splines have the smallest possible support.